Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #2464 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,12)(13,21)(14,15)(16,19)(17,23)(18,24)(20,22)(25,26,30,32)(27,31,35,39)(28,33,38,36)(29,37,40,34);(1,2)(3,4)(5,6)(7,8)(10,15)(12,16)(17,21)(20,24)(25,27,30,35)(26,31,32,39)(28,34,38,37)(29,36,40,33);(2,6)(3,7)(10,15)(12,16)(17,21)(20,24)(26,32)(27,35)(33,36)(34,37);(3,7)(4,8)(9,11,14,19)(10,16,15,12)(13,22,23,18)(17,20,21,24)(25,28,30,38)(26,33,32,36)(27,34,35,37)(29,39,40,31);(1,3)(2,4)(5,7)(6,8)(9,12,14,16)(10,11,15,19)(13,20,23,24)(17,18,21,22)(25,27)(26,31)(28,37)(29,36)(30,35)(32,39)(33,40)(34,38);(1,4)(2,3)(5,8)(6,7)(9,13,14,23)(10,17,15,21)(11,18,19,22)(12,20,16,24)(25,29,30,40)(26,34,32,37)(27,36,35,33)(28,31,38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8972831);